Gradient evaluation in Finite Volume Methods
Is there a method that is based on "coordinate transformation" to evaluate the gradient of the variables at the interface between two adjacent cells in 2D provided the coordinates of the two cell centers and two points that link the interface line? If you happen to know that, would you give me a hint? :D Thanks very much!

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Hi Yidonga, if you use finite volume,you do not need to use coordinate transformation. This should answer to you question 
leflix,
would you please introduce the reference you got the method? 
Thanks a lot! This question originally comes from a teacher's assignment in FV CFD, and I really have no idea what it means by "based on coordinate transformation theory", since from my knowledge, "coordinate transformation" is a concept from Finite Element Method, but not Finite Volume Method. Except from the method you provide, I also know there is the GreenGauss method and leastsquares method. Do you know any other method that can work good? Thanks!:D
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the method is just derivation of simple differential geometry and vector calculus. check the pdf file. But here are few references I found very comprehensive and usefull about this topic. Journal of Computational Physics 162, 411–428 (2000) Computers & Fluids 57, 225–236 (2012) Journal of Computational Physics , 228:5148–59 (2009) you have different implementations possible for this normal gradient evaluation. one which involves phi_a and phi_b and requires additional interpolation process but which does not require gradient reconstruction (ref 1 and 2). And one which avoids theses interpolations but requires gradient reconstruction. Make your choice... ;) 
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With FEM or FVM such transformation is absolutely not necessary. Quote:
In the expression I provided you do not need to reconstruct the gradient. But you need to interpolate phy_a and phy_b from cell centers values. Different implementation would avoid these interpolation but will require gradient reconstruction. See my previous post and check the book from Peric Computational Methods Fluid Dynamics. other methods to reconstruct the gradient are based on shape functions. This time it is more linked with FEM. Knowing the value of your function at given nodes (typically the vertices of your cell), you try to determine an analytical function which will interpolate these values. Then with this analytical function is is easy to derive it and to obtain the gradient on the location you need. 
Again, thanks so much for the detailed explanation! Now I'm clear that the requirement is to transform the physical domain (x,y) into the generalized coordinate system (xi,eta) and then compute gradient(U)·n x Area at interface. But I'm not sure how to write the transformation function
x=x(xi,eta) and y=y(xi,eta) or xi=(x,y) and eta=(x,y) If I can know them, I think I'll be completely clear. Thank you!:D Quote:

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http://www.erc.msstate.edu/publications/gridbook/ but as I stated it, if you use finite volume,you do not need to use such transformation. You directly work in the physical domain which is the same as computational domain. or if you absolutely want to use coordinate transformation switch to finite difference method. 
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